The \(n\times n\)  matrix polynomial \(A - \lambda E\)  is called a pencil and, additionally, it is deemed regular when \(\det (A-\lambda E)\lnot \equiv 0\)  (see chapter 12 of [1]} for more details). The finite poles and transmission zeros of any \(\mathbf {G}\in \mathbb {R}_p(\lambda )^{p \times m}\)  are rigorously defined in terms of the Smith-McMillan Form (see section 6.5.2 of [2]}). Any realization (REF )-() that is both controllable and observable is minimal (see section 6.2.2 of [2]}) and the poles of its TFM coincide with the eigenvalues of \(A\) , where \(A-\lambda I_n\)  is called the realization's pole pencil. A TFM's normal rank, which is its rank for almost all points in \({\mathbb {C}}\) , is also rigorously defined via the TFM's Smith-McMillan Form.
